Let $\pi:E\longrightarrow M$ be a vector bundle. Then we can associate a Lie groupoid $\mathsf{Gl}(E)\rightrightarrows M$ where $$\mathsf{Gl}(E):=\{E_x\stackrel{lin. isom.}{\longrightarrow} E_y: x, y\in M\}.$$ An action of a Lie groupoid $\mathsf{G}\rightrightarrows M$ on $E\stackrel{\pi}{\longrightarrow} M$ is a Lie groupoid morphism $$\mathsf{G}\longrightarrow \mathsf{Gl}(E).$$ How to define an action of a Lie groupoid on a lie algebroid? Of course, it must be an action on the underlying vector bundle plus some compatibility conditions with the anchor and the bracket on the space of sections.

I believe this is done somewhere in the literature but I couldn't find it.



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